a$.$ Prove that a DFA over the alphabet $\{0,1\}$ which recognizes the language $\{1\}$ needs to have at least $3$ states. $($Think along the lines of: I need to have atleast one start state. Now, can I get away with just that? What about the final state $?$ also how would my transitions work such that my language only accepts the string $1$ but nothing else eg empty string, $11,10$ etc$)$

b$.$ Does the DFA over the alphabet $\{1\}$ which recognizes the language $\{1\}$ also need to have at least $3$ states. Explain why or why not.